A099234 -id:A099234 - cp's OEIS frontend

A362238 Expansion of e.g.f.: 1/(1 - x*(1+x)^x).

1, 1, 2, 12, 60, 460, 3900, 39438, 456288, 5896224, 85230000, 1349017560, 23353941600, 437432418696, 8828284404576, 190867622500800, 4401749312069760, 107859517575659520, 2798352667710645120, 76636669899079699776, 2209235394261812751360

Offset: 0

Seiichi Manyama, Apr 12 2023

nonn

Cf. A202152, A362237.

Cf. A002478, A099234, A099235.

my(N=30, x='x+O('x^N)); Vec(serlaplace(1/(1-x*(1+x)^x)))

a(n) = n! * Sum_{i=0..n} Sum_{j=0..n-i} i^j * Stirling1(n-i-j,j)/(n-i-j)!.

A275277 a(n) = a(n-1) + 3a(n-2) + 3a(n-3) + a(n-4), where a(0) = a(1) = a(2) = a(3) = 1.

Original entry on oeis.org

1, 1, 1, 1, 8, 15, 43, 113, 295, 778, 2045, 5377, 14141, 37185, 97784, 257139, 676187, 1778141, 4675903, 12296026, 32334345, 85028273, 223595289, 587979169, 1546184200, 4065935847, 10692021243, 28116360553, 73936416023, 194427497258, 511277848229

Offset: 0

Clark Kimberling, Aug 11 2016

Clark Kimberling, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (1,3,3,1).

Cf. A099234.

LinearRecurrence[{1, 3, 3, 1}, {1, 1, 1, 1}, 50]
RecurrenceTable[{a[0]==a[1]==a[2]==a[3]==1,a[n]==a[n-1]+3a[n-2]+ 3a[n-3]+ a[n-4]},a,{n,30}] (* Harvey P. Dale, Apr 09 2022 *)

a(n) = a(n-1) + 3*a(n-2) + 3*a(n-3) + a(n-4), where a(0) = a(1) = a(2) = a(3) = 1.

G.f.: (-1 + 3 x^2 + 6 x^3)/(-1 + x + 3 x^2 + 3 x^3 + x^4).

A367284 G.f. satisfies A(x) = 1 + xA(x) (1 + x*A(x)^3)^3.

Original entry on oeis.org

1, 1, 4, 19, 107, 648, 4144, 27500, 187654, 1308361, 9280049, 66749995, 485741501, 3569653591, 26454406231, 197482954338, 1483619134872, 11208536870979, 85101381927454, 649017399223259, 4969510058193925, 38189305411228229, 294440263583908772

Offset: 0

Seiichi Manyama, Nov 12 2023

nonn

Cf. A002293, A099234, A364742, A367261.

a(n, s=3, t=1, u=3) = sum(k=0, n, binomial(t*k+u*(n-k)+1, k)*binomial(s*k, n-k)/(t*k+u*(n-k)+1));

If g.f. satisfies A(x) = 1 + x*A(x)^t * (1 + x*A(x)^u)^s, then a(n) = Sum_{k=0..n} binomial(t*k+u*(n-k)+1,k) * binomial(s*k,n-k) / (t*k+u*(n-k)+1).

cp's OEIS Frontend

A362238 Expansion of e.g.f.: 1/(1 - x*(1+x)^x).

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A275277 a(n) = a(n-1) + 3a(n-2) + 3a(n-3) + a(n-4), where a(0) = a(1) = a(2) = a(3) = 1.

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A367284 G.f. satisfies A(x) = 1 + xA(x) (1 + x*A(x)^3)^3.

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Author

Keywords

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PARI

Formula

cp's OEIS Frontend

A362238 Expansion of e.g.f.: 1/(1 - x*(1+x)^x).

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A275277 a(n) = a(n-1) + 3*a(n-2) + 3*a(n-3) + a(n-4), where a(0) = a(1) = a(2) = a(3) = 1.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A367284 G.f. satisfies A(x) = 1 + x*A(x) * (1 + x*A(x)^3)^3.

Views

Author

Keywords

Crossrefs

Programs

PARI

Formula

A275277 a(n) = a(n-1) + 3a(n-2) + 3a(n-3) + a(n-4), where a(0) = a(1) = a(2) = a(3) = 1.

A367284 G.f. satisfies A(x) = 1 + xA(x) (1 + x*A(x)^3)^3.